Post on 13-Apr-2018
transcript
Prepared by Kelly Wong 2009
Cointegration Test – ARDL Bounds Testing
The VAR(p) model can be rewritten in vector ECM form as:
∆𝑧𝑡 = 𝑎0 + 𝑎1𝑡𝑟𝑒𝑛𝑑 + 𝛑𝑧𝑡−1 + ∀𝑖∆𝑧𝑡−𝑖
𝜌−1
𝑖=1
+ 𝜀𝑡
where
∆ = 1 – L is the difference operator,
zt = f(yt, xt)
we now partition the long-run multiplier matrix 𝜋 conformably with zt = (yt, x’t)’ as
𝜋 = 𝜋𝑦𝑦 𝜋𝑦𝑥
𝜋𝑥𝑦 𝜋𝑥𝑥
Under the assumption 1, 3, and 4 (see Pesaran et al. 2001), 𝜋 has rank r and is given by
𝜋 = 𝜋𝑦𝑦 𝜋𝑦𝑥
0 𝜋𝑥𝑥
Consequently, the conditional ECM may be written as following:
∆𝑦𝑡 = 𝑎0 + 𝑎1𝑡𝑟𝑒𝑛𝑑 + 𝛑𝑦𝑦 𝑦𝑡−1 + 𝛑𝑦𝑥 .𝑥𝑥𝑡−1 + ∀𝑖∆𝑧𝑡−𝑖
𝜌−1
𝑖=1
+𝑤 ′∆𝑥𝑡 + 𝜀𝑡
If the 𝜋𝑦𝑦 ≠ 0 and 𝜋𝑦𝑥 .𝑥 = 0′ , the yt is (trend) stationary or I(0), whatever the value r.
Consequently, the differenced variable ∆𝑦𝑡 depends only on its own lagged level yt-1 in the
conditional ECM. Second, if 𝜋𝑦𝑦 = 0 , the ∆𝑦𝑡 depends only on the lagged level xt-1 in the
conditional ECM model. Therefore, in order to test for the absence of level effects in the
conditional ECM model and more crucially, the absence of a level relationship between yt and xt,
the emphasis in this approach is a test of the joint hypothesis the 𝜋𝑦𝑦 = 0 and 𝜋𝑦𝑥 .𝑥 = 0′ in the
above model.
According to Pesaran et al. (2001), there are 5 cases provided for testing the cointegrating bound
test:
Case 1: (no intercepts; no trends) a0 and a1 = 0.
Case 2: (restricted intercepts; no trends) a0 = - (𝜋𝑦𝑦 , 𝜋𝑦𝑥 .𝑥)𝜇 and a1 = 0.
Prepared by Kelly Wong 2009
Case 3: (unrestricted intercepts; no trends) 𝑎0 ≠ 0 and 𝑎1 = 0.
Case 4: (unrestricted intercepts; restricted trends) 𝑎0 ≠ 0 and a1 = - (𝜋𝑦𝑦 , 𝜋𝑦𝑥 .𝑥)𝜇
Case 5: (unrestricted intercepts; unrestricted trends) 𝑎0 ≠ 0 and 𝑎1 ≠ 0
______________________________________________________________________________
If the Wald F-statistic fall above the upper critical value – cointegrated
If the Wald F-statistic falls between the lower bound and upper bound critical value –
inconclusive
If the Wald F-statistic falls below the lower bound critical value – no cointegration
______________________________________________________________________________
Example: Based on the theory, the financial development will be affected the national poverty.
So, it is likely to investigate whether the financial development have decreases the Malaysian
poverty gap. Testing the regression model - RPCC = f(RGDPC, UN, WAGE, TAX, LL) by using
the secondary data from poverty and fd.xls file.
Rewriting the ARDL Bound Cointegration Test model:
∆𝑅𝑃𝐶𝐶𝑡 = 𝑐 + 𝛽1𝑅𝐺𝐷𝑃𝐶𝑡−1 + 𝛽2𝑈𝑁𝑡−1 + 𝛽3𝑊𝐴𝐺𝐸𝑡−1 + 𝛽4𝑇𝐴𝑋𝑡−1 + 𝛽5𝐿𝐿𝑡−1 +
𝛼1𝑖
𝑝
𝑖=1
∆𝑅𝑃𝐶𝐶𝑡−𝑖 + 𝛼2𝑖
𝑝
𝑖=0
∆𝑈𝑁𝑡−𝑖 + 𝛼3𝑖
𝑝
𝑖=0
∆𝑊𝐴𝐺𝐸𝑡−𝑖 +
𝛼4𝑖
𝑝
𝑖=0
∆𝑇𝐴𝑋𝑡−𝑖 + 𝛼5𝑖
𝑝
𝑖=0
∆𝐿𝐿𝑡−𝑖 + 𝛾𝐷𝑈𝑀
where RPCC = real Consumption Per capita
C = constant
RGDPC = real GDP Per capita
UN = unemployment
WAGE = Wages
TAX = Individual Tax
LL = Liquid Liability
p = optimum lag length uses in model
Prepared by Kelly Wong 2009
DUM = 1 for crisis and 0 for otherwise; the crisis is refer to the Malaysian oil
crisis at 1973, 1974, 1980 and 1981; commodities crisis at 1985 to 1986; and 1997/98 for
Asian financial Crisis.
Firstly, we should examine this Bound Testing model start from the higher lag length. For
example, in this model have only 38 observations and consider as small size sample time series
model. In order to avoid the overparameter problem, this example start with the maximum lag
order 2 (maximum lag) and then reduces to maximum lag 1:
Definition for the Eviews code:
d = change
-1 = lag one variable = time T – 1
c = constant term
Go to Quick – Estimate Equation –
Type in the model.
Prepared by Kelly Wong 2009
Dependent Variable: D(RPCC)
Method: Least Squares
Sample (adjusted): 1973 2004
Included observations: 32 after adjustments Variable Coefficient Std. Error t-Statistic Prob. C 13.77331 2.126913 6.475728 0.0003
RPCC(-1) 0.102209 0.124537 0.820713 0.4389
RGDPC(-1) -2.033942 0.344902 -5.897155 0.0006
UN(-1) -0.089575 0.026223 -3.415877 0.0112
WAGE(-1) -0.520555 0.080194 -6.491173 0.0003
TAX(-1) 0.563704 0.096699 5.829448 0.0006
LL(-1) -0.050592 0.056886 -0.889364 0.4033
D(RPCC(-1)) -0.618619 0.221448 -2.793518 0.0268
D(RPCC(-2)) -0.508969 0.173976 -2.925506 0.0222
D(RGDPC) 0.558362 0.143865 3.881151 0.0060
D(RGDPC(-1)) 1.912944 0.372013 5.142141 0.0013
D(RGDPC(-2)) 0.683437 0.278168 2.456918 0.0437
D(UN) -0.110758 0.045845 -2.415939 0.0464
D(UN(-1)) -0.015647 0.031613 -0.494933 0.6358
D(UN(-2)) -0.026159 0.024813 -1.054212 0.3268
D(WAGE) -0.179064 0.035233 -5.082250 0.0014
D(WAGE(-1)) 0.124112 0.048005 2.585401 0.0362
D(WAGE(-2)) -0.067168 0.022898 -2.933300 0.0219
D(TAX) 0.166026 0.044277 3.749698 0.0072
D(TAX(-1)) -0.216132 0.044101 -4.900844 0.0018
D(TAX(-2)) -0.069359 0.028300 -2.450836 0.0441
D(LL) -0.107055 0.033535 -3.192336 0.0152
D(LL(-1)) -0.103247 0.028797 -3.585338 0.0089
D(LL(-2)) -0.022240 0.021554 -1.031829 0.3365
DUM -0.008776 0.014612 -0.600579 0.5670 R-squared 0.992074 Mean dependent var 0.031780
Adjusted R-squared 0.964899 S.D. dependent var 0.056345
S.E. of regression 0.010556 Akaike info criterion -6.221481
Sum squared resid 0.000780 Schwarz criterion -5.076375
Log likelihood 124.5437 Hannan-Quinn criter. -5.841910
F-statistic 36.50675 Durbin-Watson stat 2.664106
Prob(F-statistic) 0.000030
Diagnostic Checking for Serial LM test – Click ―View‖ – ―Residual Test‖ – ―Serial Correlation
LM Test‖:
Breusch-Godfrey Serial Correlation LM Test: F-statistic 1.264529 Prob. F(2,5) 0.3594
Obs*R-squared 10.74900 Prob. Chi-Square(2) 0.0046
Breusch-Godfrey Serial Correlation LM Test: F-statistic 0.687702 Prob. F(4,3) 0.6471
Obs*R-squared 15.30670 Prob. Chi-Square(4) 0.0041
Prepared by Kelly Wong 2009
After the Lag 2 model, we have attempts to reduce the lag to be lag 1 models. The empirical
result is shows as following:
Dependent Variable: D(RPCC)
Method: Least Squares
Sample (adjusted): 1972 2004
Included observations: 33 after adjustments Variable Coefficient Std. Error t-Statistic Prob. C 6.078979 1.778360 3.418306 0.0042
RPCC(-1) -0.113193 0.180452 -0.627273 0.5406
RGDPC(-1) -0.753554 0.275372 -2.736499 0.0161
UN(-1) -0.036190 0.039083 -0.925960 0.3702
WAGE(-1) -0.232102 0.066913 -3.468741 0.0038
TAX(-1) 0.238435 0.076828 3.103505 0.0078
LL(-1) -0.002917 0.065584 -0.044480 0.9652
D(RPCC(-1)) 0.047289 0.192419 0.245759 0.8094
D(RGDPC) 0.808263 0.231515 3.491185 0.0036
D(RGDPC(-1)) 0.739431 0.295086 2.505812 0.0252
D(UN) -0.025586 0.065172 -0.392589 0.7005
D(UN(-1)) 0.033920 0.042539 0.797381 0.4385
D(WAGE) -0.124827 0.057956 -2.153823 0.0492
D(WAGE(-1)) -0.000677 0.041739 -0.016213 0.9873
D(TAX) 0.074923 0.049829 1.503584 0.1549
D(TAX(-1)) -0.068402 0.044450 -1.538861 0.1461
D(LL) -0.057921 0.043217 -1.340219 0.2015
D(LL(-1)) -0.060747 0.031568 -1.924284 0.0749
DUM -0.013055 0.018564 -0.703274 0.4934 R-squared 0.939614 Mean dependent var 0.031590
Adjusted R-squared 0.861974 S.D. dependent var 0.055469
S.E. of regression 0.020608 Akaike info criterion -4.632248
Sum squared resid 0.005945 Schwarz criterion -3.770623
Log likelihood 95.43210 Hannan-Quinn criter. -4.342337
F-statistic 12.10229 Durbin-Watson stat 2.400846
Prob(F-statistic) 0.000012
Series LM Test:
Breusch-Godfrey Serial Correlation LM Test: F-statistic 0.924306 Prob. F(2,12) 0.4233
Obs*R-squared 4.405075 Prob. Chi-Square(2) 0.1105
Breusch-Godfrey Serial Correlation LM Test: F-statistic 1.240701 Prob. F(4,10) 0.3544
Obs*R-squared 10.94531 Prob. Chi-Square(4) 0.0272
Prepared by Kelly Wong 2009
After the Eviews output, we need to prepaid a table to record the AIC, SBC, and Serial LM test.
For example:
Table Exp.1. Selected Maximum Lag Base on difference Method
P AIC SBC )2(SCx )4(SCx
2 -6.2215 -5.0764 10.74*** 15.30***
1 -4.6322 -3.7706 4.4051 10.945**
Note: p is the lag order of the underlying VAR model for the conditional ECM ( ), with zero restrictions on
the coefficients of lagged changes in the independent variables. AICp = (-2l / T) + (2k / T) and SBCp = (-
2l / T) + (k * logT / T) denote Akaike’s and Schwarz’s Bayesian Information Criteria for a given lag order
p, where l is the maximized log-likelihood value of the model, k is the number of freely estimated
coefficients and T is the sample size. The AIC and SBC are often used in model selection for non-nested
alternatives—lowest values of the AIC and SBC are preferred (refer to Eviews Users Guide 4.0, pp. 279).
Xsc (1) and Xsc (4) are LM statistics for testing no residual serial correlation against orders 2 and 4. The
symbols ***, and ** denote significance at 0.01, and 0.05 levels, respectively.
The above estimated results showed that serial correlation problem exists in all lag models.
Hence, we should reconsider other methods for chosen the optimum lag length such as Hendry’s
General to specific model. Again, we start the model at maximum lag 2. After that, delete the
higher insignificant lag for the changes variables. For example: the D(LL(-2)) should be drop out
from the model first and then following by D(UN(-2)) and D(UN(-1)). At the end, the result
should be as follow:
Prepared by Kelly Wong 2009
Dependent Variable: D(RPCC)
Method: Least Squares
Sample (adjusted): 1973 2004
Included observations: 32 after adjustments Variable Coefficient Std. Error t-Statistic Prob. C 12.61967 1.731161 7.289714 0.0000
RPCC(-1) 0.055598 0.104682 0.531110 0.6069
RGDPC(-1) -1.817123 0.270645 -6.714037 0.0001
UN(-1) -0.074733 0.021336 -3.502689 0.0057
WAGE(-1) -0.469390 0.062083 -7.560639 0.0000
TAX(-1) 0.523991 0.082461 6.354384 0.0001
LL(-1) -0.087001 0.042415 -2.051171 0.0674
D(RPCC(-1)) -0.513899 0.177106 -2.901654 0.0158
D(RPCC(-2)) -0.442000 0.151922 -2.909383 0.0156
D(RGDPC) 0.634967 0.115742 5.486054 0.0003
D(RGDPC(-1)) 1.749137 0.290044 6.030600 0.0001
D(RGDPC(-2)) 0.609775 0.226866 2.687824 0.0228
D(UN) -0.081451 0.034959 -2.329896 0.0421
D(WAGE) -0.166054 0.029206 -5.685552 0.0002
D(WAGE(-1)) 0.103437 0.039845 2.595963 0.0267
D(WAGE(-2)) -0.068902 0.020769 -3.317622 0.0078
D(TAX) 0.152515 0.039787 3.833305 0.0033
D(TAX(-1)) -0.203931 0.038001 -5.366529 0.0003
D(TAX(-2)) -0.072995 0.025175 -2.899487 0.0158
D(LL) -0.111729 0.030455 -3.668680 0.0043
D(LL(-1)) -0.075093 0.016830 -4.461815 0.0012
DUM -0.010083 0.013219 -0.762732 0.4632 R-squared 0.990009 Mean dependent var 0.031780
Adjusted R-squared 0.969028 S.D. dependent var 0.056345
S.E. of regression 0.009916 Akaike info criterion -6.177457
Sum squared resid 0.000983 Schwarz criterion -5.169763
Log likelihood 120.8393 Hannan-Quinn criter. -5.843435
F-statistic 47.18594 Durbin-Watson stat 2.520468
Prob(F-statistic) 0.000000
Breusch-Godfrey Serial Correlation LM Test: F-statistic 0.840791 Prob. F(2,8) 0.4662
Obs*R-squared 5.558040 Prob. Chi-Square(2) 0.0621
Breusch-Godfrey Serial Correlation LM Test: F-statistic 0.597855 Prob. F(4,6) 0.6781
Obs*R-squared 9.119481 Prob. Chi-Square(4) 0.0582
After the Hendry’s
method applied, the
problems of serial
correlation were
overcome.
After the Hendry’s
method applied, the
problems of serial
correlation were not
significant at 5%
significance level.
Prepared by Kelly Wong 2009
After specified the optimum lag model, then we should process to the ARDL Cointegration
Bound Testing. Firstly, we need to understand the code used in the Eviews program represented.
Second, according to Pesaran et al. (2001), if the coefficients between lag one variables are
jointly fall above the upper bound critical value, then indicates that these estimated variables
exists a long-run cointegration relationship. In order to testing this hypothesis, we need estimate
the coefficient RPCC(-1) = RGDPC(-1) = UN(-1) = WAGE(-1) = TAX(-1) = LL(-1) = 0;
rewriting in the Eviews code as c(2) = c(3) = c(4) = c(5) = c(6) = c(7) = 0. Hence, now should
click on the ―view‖ button – ―Coefficient Tests‖ – ―Wald coefficient restrictions‖ .
Dependent Variable: D(RPCC)
Method: Least Squares
Sample (adjusted): 1973 2004
Included observations: 32 after adjustments Variable Coefficient Std. Error t-Statistic Prob. C 12.61967 1.731161 7.289714 0.0000
RPCC(-1) 0.055598 0.104682 0.531110 0.6069
RGDPC(-1) -1.817123 0.270645 -6.714037 0.0001
UN(-1) -0.074733 0.021336 -3.502689 0.0057
WAGE(-1) -0.469390 0.062083 -7.560639 0.0000
TAX(-1) 0.523991 0.082461 6.354384 0.0001
LL(-1) -0.087001 0.042415 -2.051171 0.0674
D(RPCC(-1)) -0.513899 0.177106 -2.901654 0.0158
D(RPCC(-2)) -0.442000 0.151922 -2.909383 0.0156
D(RGDPC) 0.634967 0.115742 5.486054 0.0003
D(RGDPC(-1)) 1.749137 0.290044 6.030600 0.0001
D(RGDPC(-2)) 0.609775 0.226866 2.687824 0.0228
D(UN) -0.081451 0.034959 -2.329896 0.0421
D(WAGE) -0.166054 0.029206 -5.685552 0.0002
D(WAGE(-1)) 0.103437 0.039845 2.595963 0.0267
D(WAGE(-2)) -0.068902 0.020769 -3.317622 0.0078
D(TAX) 0.152515 0.039787 3.833305 0.0033
D(TAX(-1)) -0.203931 0.038001 -5.366529 0.0003
D(TAX(-2)) -0.072995 0.025175 -2.899487 0.0158
D(LL) -0.111729 0.030455 -3.668680 0.0043
D(LL(-1)) -0.075093 0.016830 -4.461815 0.0012
DUM -0.010083 0.013219 -0.762732 0.4632 R-squared 0.990009 Mean dependent var 0.031780
Adjusted R-squared 0.969028 S.D. dependent var 0.056345
S.E. of regression 0.009916 Akaike info criterion -6.177457
Sum squared resid 0.000983 Schwarz criterion -5.169763
Log likelihood 120.8393 Hannan-Quinn criter. -5.843435
F-statistic 47.18594 Durbin-Watson stat 2.520468
Prob(F-statistic) 0.000000
C(1)
C(2)
C(3)
C(4)
C(5)
C(6)
C(7)
C(8)
c(9)
C(10)
C(11)
C(12)
C(13)
C(14)
C(15)
C(16)
C(17)
C(18)
C(19)
C(20)
C(21)
C(22)
Prepared by Kelly Wong 2009
Wald Test:
Equation: Untitled Test Statistic Value df Probability F-statistic 14.63692 (6, 10) 0.0002
Chi-square 87.82149 6 0.0000
After click on the Wald –
Coefficient Restrictions
test, the Eviews Program
will occurs a empty
windows for put in the
null hypothesis testing. In
our Example, we should
type in the code as
showed in the figure.
Compare this F-statistic with the
Narayan (2005) Critical value based
on the case you chose before, if your
sample size is relative small.
Prepared by Kelly Wong 2009
Prepare a table to record the information, such as:
Table Exp.2. F-statistics for testing the existence of levels Poverty equation
Model F-statistic
Model 1: RPCC = f (RGDPC, UN, WAGE, TAX, LL)
14.6369
Narayan (2005) k = 6, n=35
Critical Value Lower bound Upper bound
1%
5%
10%
4.016
2.864
2.387
5.797
4.324
3.671
Notes: *, **, and *** denote significant at 10%, 5%, and 1% significance level, respectively. Critical
values are cited from Narayan (2005) (Table Case III: Unrestricted intercept and no trend; pg. 1988).
In this Example, the model shows that the financial development and other determinant variables
are strongly cointegrated with poverty in Malaysia. The results showed that the F-statistic
compute by Wald Test is highly significant at 1% significance level.
Prepared by Kelly Wong 2009
Using Eviews to construct an ARDL Bound Test
1. These criteria are suggested for author to select the optimum lag in the ARDL modeling,
namely
1. Akaike Information Criterion
2. Schwarz Bayesian Criterion
3. General to specific model
For Example: Our regression model – RPCC = f(RGDPC, UN, WAGE, TAX, LL) (See the file poverty
and fd) and the selected optimum lag length is (1, 1, 0, 0, 0, 0)
ARDL Model – Equation (1):
it
r
i
tit
q
i
tit
p
i
t UNRGDPCRPCCconstRPCC0
,3
0
,2
1
,1
ttit
v
i
tit
u
i
tit
s
i
t DUMLLTAXWAGE
,7
0
,6
0
,5
0
,4
where
RPCC = real Consumption Per capita
Const = constant
RGDPC = real GDP Per capita
UN = unemployment
WAGE = Wages
TAX = Individual Tax
LL = Liquid Liability
DUM = 1 for crisis and 0 for otherwise; the crisis is refer to the Malaysian oil crisis at 1973,
1974, 1980 and 1981; commodities crisis at 1985 to 1986; and 1997/98 for Asian financial
Crisis.
p, q, r, s, u, v = optimum lag length uses in model
t = residual
Such based on the AIC or SBC criteria, the selected lag length for this model (p, q, r, s, u, v) is (1, 1,
0, 0, 0, 0). This can use Microfit or Rats programming code to obtain the optimum lag base on
such listed criteria.
Prepared by Kelly Wong 2009
2. After selected lag length, using Eviews to estimate the Long-run OLS. The Eviews output
is showed as following:
Dependent Variable: RPCC
Method: Least Squares
Date: 06/28/09 Time: 21:23
Sample (adjusted): 1971 2004
Included observations: 34 after adjustments
Variable Coefficient Std. Error t-Statistic Prob.
C 1.380580 1.142938 1.207922 0.2384
RPCC(-1) 0.688508 0.175616 3.920539 0.0006
RGDPC 0.981537 0.179936 5.454926 0.0000
RGDPC(-1) -0.866497 0.237380 -3.650252 0.0012
UN 0.010227 0.036818 0.277762 0.7835
WAGE -0.019567 0.047700 -0.410202 0.6852
TAX 0.071253 0.047672 1.494647 0.1475
LL -0.095664 0.036619 -2.612439 0.0150
DUM 0.015761 0.016593 0.949891 0.3513
R-squared 0.992724 Mean dependent var 7.949910
Adjusted R-squared 0.990396 S.D. dependent var 0.307982
S.E. of regression 0.030182 Akaike info criterion -3.941225
Sum squared resid 0.022774 Schwarz criterion -3.537189
Log likelihood 76.00083 Hannan-Quinn criter. -3.803437
F-statistic 426.3952 Durbin-Watson stat 1.789455
Prob(F-statistic) 0.000000
3. After the ARDL model, we have using the ―Wald Test‖ to compute the long run
elasticities and it standard error.
According to Pesaran et al. (2001) the Long run elasticities should compute as follow:
p
i
t
q
i
t
RGDPCforesElasticiti
,1
,2
1
__
= Sum of the independent coefficients (RGDPC)
1 – Sum of the dependent coefficients
The coefficient for the
variables in the Eviews
output is shows as:
c(1)
c(2)
c(3)
c(4)
c(5)
c(6)
c(7)
c(8)
c(9)
Prepared by Kelly Wong 2009
4. Go to ―View‖ – ―Coefficient Test‖ – ―Wald Test‖
Eviews Output:
Wald Test:
Equation: Untitled
Test Statistic Value df Probability
F-statistic 0.507750 (1, 25) 0.4827
Chi-square 0.507750 1 0.4761
Null Hypothesis Summary:
Normalized Restriction (= 0) Value Std. Err.
(C(3) + C(4)) / (1 - C(2)) 0.369318 0.518293
Delta method computed using analytic derivatives.
For others variables elasticities are showed as: constant (4.4322), RGDPC (0.36932), UN (0.032831),
WAGE (-0.062815), TAX (0.22875), LL (-0.30712), DUM (0.050599).
Type in the code:
(c(3)+c(4))/(1-c(2))=0
In the Wald Test windows
The value of elasticities is
shows as 0.369318. The
standard error is
0.518293. However, the t-
statistic need compute by
the user, where t-stat =
coefficient / std. Err.
The probability 0.4827
also represent as p-value
for the computed
elasticities.
Prepared by Kelly Wong 2009
5. After computed the all long-run elasticities, we need proceed into short-run Error
correction Model.
a. Firstly, compute the values of Error Correction Term (ECT). Based on the
knowledge the ECT is represent as a long-run steady point for the model or more
statistically the ECT is a residual from long-run cointegration model. Long-run
Cointegration Model (Equation 2):
tp
i
t
s
i
t
tp
i
t
r
i
t
tp
i
t
q
i
t
p
i
t
t WAGEUNRGDPCconst
RPCC
1
,1
0
,4
1
,1
0
,3
1
,1
0
,2
,1 1111
tp
i
t
tp
i
t
v
i
t
tp
i
t
u
i
t
ECTDUMLLTAX
1
,1
7
1
,1
0
,6
1
,1
0
,5
111
(2)
After some mathematical adjustment, the Error correction term equation is shows as:
tp
i
t
s
i
t
tp
i
t
r
i
t
tp
i
t
q
i
t
t WAGEUNRGDPCRPCCECT
1
,1
0
,4
1
,1
0
,3
1
,1
0
,2
111
tp
i
t
v
i
t
tp
i
t
u
i
t
LLTAX
1
,1
0
,6
1
,1
0
,5
11
(3)
Therefore, based on the previous example model and using the calculated elasticities, the Long-run
Cointegrated Model is shows as following:
RPCC = 4.4322 + 0.36932 RGDPC + 0.032831 UN – 0.062815 WAGE + 0.22875 TAX – 0.30712 LL +
0.050599 DUM
Hence, the ECT equation shows as:
ECT = RPCC– 0.36932*RGDPC – 0.032831*UN + 0.062815*WAGE – 0.22875*TAX + 0.30712*LL
So, generate this ECT equation in the Eviews before the Short-run dynamic model.
Prepared by Kelly Wong 2009
b. Type in the ECT equation on the upper blank box of Eviews and then “Enter”.
6. After generated the ECT series, now we have go to “Quick” – “Estimate Equation” – choose the
method “TSLS”.
Type in this equation on the top blank
box:
ECT = RPCC – 0.36932*RGDPC –
0.032831*UN + 0.062815*WAGE –
0.22875*TAX + 0.30712*LL
Furthermore, press the “Enter” and the
ECT will shows in the workfile windows.
Prepared by Kelly Wong 2009
7. Now the Eviews shows you two boxes, one is for Equation specification and other one is for
instrument list.
a. The Equation Specification is refer to the Short-run dynamic model, which is:
it
q
i
tit
p
i
ttt RGDPCRPCCECTconstRPCC1
0
,2
1
1
,11
it
u
i
tit
s
i
tit
r
i
t TAXWAGEUN1
0
,5
1
0
,4
1
0
,3
ttit
v
i
t DUMLL
,7
1
0
,6
For our Example:
The Equation Specification code is follows:
D(RPCC) C ECT(-1) D(RGDPC) D(UN) D(WAGE)
D(TAX) D(LL) DUM
b. The instrument list refers to the endogenous for ECT models. For our Example, the Eviews code to
represent the exogenous variables for our ECT model is:
C RPCC(-1) RGDPC RGDPC(-1) UN WAGE TAX
LL
Type in this equation on the
Equation Specification box:
D(RPCC) C ECT(-1) D(RGDPC)
D(UN) D(WAGE) D(TAX) D(LL)
DUM
Furthermore, type in the
instrument list:
C RPCC(-1) RGDPC RGDPC(-1) UN
WAGE TAX LL
Prepared by Kelly Wong 2009
After that click “Options” and tick that “Heteroskedasticity consistent coefficient covariance” and
“Newey-West” and then click ok.
8. After that you should get the Short-run dynamic results as follows:
Dependent Variable: D(RPCC)
Method: Two-Stage Least Squares
Date: 06/29/09 Time: 10:19
Sample (adjusted): 1971 2004
Included observations: 34 after adjustments
Newey-West HAC Standard Errors & Covariance (lag truncation=3)
Instrument list: C RPCC(-1) RGDPC RGDPC(-1) UN WAGE TAX LL
Variable Coefficient Std. Error t-Statistic Prob.
C 1.380587 1.160771 1.189371 0.2450
ECT(-1) -0.311494 0.259603 -1.199883 0.2410
D(RGDPC) 0.981566 0.410089 2.393546 0.0242
D(UN) 0.010237 0.146174 0.070031 0.9447
D(WAGE) -0.019561 0.058823 -0.332532 0.7422
D(TAX) 0.071255 0.046388 1.536072 0.1366
D(LL) -0.095667 0.049627 -1.927704 0.0649
DUM 0.015762 0.018952 0.831679 0.4132
R-squared 0.768700 Mean dependent var 0.031574
Adjusted R-squared 0.706427 S.D. dependent var 0.054622
S.E. of regression 0.029595 Sum squared resid 0.022773
F-statistic 12.34394 Durbin-Watson stat 1.789467
Prob(F-statistic) 0.000001 Second-Stage SSR 0.022774
Click on “Options” and tick
the box of
“Heteroskedasticity
consistent coefficient
covariance” and “Newey-
West”.
And then click “ok”.
Prepared by Kelly Wong 2009
As compared to the Microfit Output:
Error Correction Representation for the Selected ARDL Model
ARDL(1,1,0,0,0,0) selected based on Schwarz Bayesian Criterion
*******************************************************************************
Dependent variable is dRPCC
34 observations used for estimation from 1971 to 2004
*******************************************************************************
Regressor Coefficient Standard Error T-Ratio[Prob]
dRGDPC .98154 .17994 5.4549[.000]
dUN .010227 .036818 .27776[.783]
dWAGE -.019567 .047700 -.41020[.685]
dTAX .071253 .047672 1.4946[.147]
dLL -.095664 .036619 -2.6124[.015]
dINPT 1.3806 1.1429 1.2079[.238]
dDUM .015761 .016593 .94989[.351]
ecm(-1) -.31149 .17562 -1.7737[.088]
*******************************************************************************
List of additional temporary variables created:
dRPCC = RPCC-RPCC(-1)
dRGDPC = RGDPC-RGDPC(-1)
dUN = UN-UN(-1)
dWAGE = WAGE-WAGE(-1)
dTAX = TAX-TAX(-1)
dLL = LL-LL(-1)
dINPT = INPT-INPT(-1)
dDUM = DUM-DUM(-1)
ecm = RPCC -.36932*RGDPC -.032831*UN + .062815*WAGE -.22875*TAX + .30
712*LL -4.4322*INPT -.050599*DUM
*******************************************************************************
R-Squared .76870 R-Bar-Squared .69468
S.E. of Regression .030182 F-stat. F( 7, 26) 11.8689[.000]
Mean of Dependent Variable .031574 S.D. of Dependent Variable .054622
Residual Sum of Squares .022774 Equation Log-likelihood 76.0008
Akaike Info. Criterion 67.0008 Schwarz Bayesian Criterion 60.1322
DW-statistic 1.7895
*******************************************************************************
R-Squared and R-Bar-Squared measures refer to the dependent variable
dRPCC and in cases where the error correction model is highly
restricted, these measures could become negative.